Finite Group Databasebeta

← #28499403

Subgroups — #28499403

Order 15104.

402 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427363C211yesmaximal, characteristic
2#21424172C211yesmaximal, characteristic
3#21422427C211yesmaximal, characteristic
4#21422121C211yesmaximal, characteristic
5#21422119C211yesmaximal, characteristic
6#21422117C211yesmaximal, characteristic
7C59×#1351C211yesmaximal, characteristic
8C2×Q8×Dic59C411yescharacteristic, complemented
9C2×#14273683C2211yescharacteristic
10#16837456—21no—
11#16837342C411yescharacteristic, complemented
12#16837338C2211yescharacteristic
13#16837336—21no—
14#16837325—21no—
15#16837321C411yescharacteristic, complemented
16#16837263—21no—
17#16837253—21no—
18#16837253C411yescharacteristic, complemented
19C22⋊C4×Dic59—21no—
20#16837169C2211yescharacteristic
21#16837168—21no—
22#16837127—21no—
23#16837126C2211yescharacteristic
24C59×#393C2211yescharacteristic
25C59×#342C2211yescharacteristic
26C59×#323C2211yescharacteristic
27C23×Dic59D411yescharacteristic
28Q8×Dic59—41no—
29Q8×Dic59—41no—
30#14273683—41no—
31#14273683—41no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59C2×C411yescharacteristic
34C2×Dic59⋊C4—21no—
35C2×Dic59⋊C4—21no—
36C2×Dic59⋊C4—21no—
37C2×Dic59⋊C4C2×C411yescharacteristic
38C2×Dic59⋊C4D411yescharacteristic
39C2×C22⋊Dic59—21no—
40C2×C22⋊Dic59—21no—
41C2×C22⋊Dic59D411yescharacteristic
42C2×C4×Dic59—21no—
43C2×C4×Dic59—21no—
44C2×C4×Dic59C2×C411yescharacteristic
45C2×C4×Dic59D411yescharacteristic
46Dic59⋊C8—41no—
47C22⋊(C59⋊C8)—21no—
48#14273637—41no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C59⋊#95—41no—
53C59⋊#95—41no—
54C59⋊#95C2×C411yescharacteristic
55C22×Q8×C59C2×C411yescharacteristic
56C2×C22⋊C4×C59—21no—
57C2×C22⋊C4×C59—21no—
58C2×C22⋊C4×C59C2311yescharacteristic
59C22⋊C8×C59—21no—
60#95×C59—21no—
61#95×C59C2×C411yescharacteristic
62C24×C59C2×D411yescharacteristic
63C2×Q8×C59—42no—
64C2×Q8×C59—41no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—41no—
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×Dic59—21no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—41no—
77C22×Dic59—41no—
78C22×Dic59C22⋊C411yescharacteristic
79C22×Dic59C22⋊C411yescharacteristic
80C22×Dic59C22⋊C411yescharacteristic
81C22×Dic59C22⋊C411yescharacteristic
82C2×C8×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—42no—
85C22⋊C4×C59—41no—
86C22⋊C4×C59—41no—
87C22⋊Dic59—41no—
88C22⋊Dic59—42no—
89C22⋊Dic59—41no—
90C22⋊Dic59—41no—
91C4⋊Dic59—42no—
92C4⋊Dic59—41no—
93Dic59⋊C4—41no—
94Dic59⋊C4—41no—
95Dic59⋊C4—42no—
96Dic59⋊C4—42no—
97Dic59⋊C4—42no—
98C4×Dic59—21no—
99C4×Dic59—21no—
100C4×Dic59—21no—
101C4×Dic59—21no—
102C4×Dic59—42no—
103C4×Dic59—41no—
104C4×Dic59—41no—
105C2×C59⋊C8—41no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59C2×C22⋊C411yescharacteristic
111C23×C59#13711yescharacteristic
112C23×C59#13711yescharacteristic
113Q8×C59—42no—
114Q8×C59—42no—
115C2×C4×C59—22no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—81no—
126C2×C4×C59—81no—
127C2×C4×C59—81no—
128C2×Dic59—21no—
129C2×Dic59—21no—
130C2×Dic59—22no—
131C2×Dic59—21no—
132C2×Dic59—21no—
133C2×Dic59—21no—
134C2×Dic59—21no—
135C2×Dic59—21no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—81no—
148C2×Dic59—81no—
149C2×Dic59—81no—
150C2×Dic59#9911yescharacteristic
151C2×Dic59#9911yescharacteristic
152C8×C59—81no—
153C59⋊C8—81no—
154#11810—591nomaximal, Sylow
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—41no—
160C22×C59—41no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—81no—
164C22×C59C2×#9911yescharacteristic
165C22×C59#41911yescharacteristic
166C22×C59#42111yescharacteristic
167C4×C59—42no—
168C4×C59—41no—
169C4×C59—41no—
170C4×C59—81no—
171C4×C59—161no—
172Dic59—21no—
173Dic59—21no—
174Dic59—42no—
175Dic59—41no—
176Dic59—41no—
177Dic59—41no—
178Dic59—81no—
179Dic59—161no—
180#2396—591no—
181#1820—591no—
182#1464—591no—
183#1352—591no—
184#1352—591no—
185#1351—591no—
186#1351D5911yescharacteristic
187C2×C59—21no—
188C2×C59—21no—
189C2×C59—41no—
190C2×C59—41no—
191C2×C59#151111yescharacteristic
192C2×C59#212011yescharacteristic
193C2×C59#212611yescharacteristic
194C2×C4×Q8—591no—
195C2×#117—591no—
196#432—1181no—
197#400—591no—
198#399—591no—
199#398—1181no—
200#396—1181no—
201#395—591no—
202#393C2×D5911yescharacteristic
203#381—1181no—
204C4×C4⋊C4—591no—
205C4×C4⋊C4—1181no—
206C4×C22⋊C4—1181no—
207#342—591no—
208#342—1181no—
209#342C2×D5911yescharacteristic
210#324—1181no—
211#323—591no—
212#323C2×D5911yescharacteristic
213C59#1181011yescharacteristic, complemented, Sylow
214C22×Q8C4×D5911yescharacteristic
215C23×C4—591no—
216C4×Q8—2361no—
217C4×Q8—2361no—
218#117—2361no—
219#117—2361no—
220C2×C4⋊C4—591no—
221C2×C4⋊C4—591no—
222C2×C4⋊C4—591no—
223C2×C4⋊C4—1181no—
224C2×C4⋊C4—1181no—
225C2×C4⋊C4—1181no—
226C2×C4⋊C4—1181no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—591no—
230C2×C22⋊C4—1181no—
231C2×C22⋊C4—1181no—
232C2×C22⋊C4C22×D5911yescharacteristic
233C2×C42—591no—
234C2×C42—591no—
235C2×C42—1181no—
236C2×C42—1181no—
237C4⋊C8—2361no—
238C22⋊C8—21no—
239C22⋊C8—1181no—
240#97—2361no—
241#95—21no—
242#95—591no—
243#95—1181no—
244#95—1181no—
245#95—1181no—
246#95—2361no—
247#95—2361no—
248#95C4×D5911yescharacteristic
249C24D4×D5911yescharacteristic
250C2×Q8—42no—
251C2×Q8—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—591no—
256C22×C4—591no—
257C22×C4—591no—
258C22×C4—591no—
259C22×C4—1181no—
260C22×C4—1181no—
261C22×C4—1181no—
262C22×C4—1181no—
263C22×C4—1181no—
264C22×C4—1181no—
265C22×C4—2361no—
266C22×C4—2361no—
267C22×C4C2×C4×D5911yescharacteristic
268C22×C4D4×D5911yescharacteristic
269C2×C8—41no—
270C2×C8—2361no—
271C4⋊C4—2361no—
272C4⋊C4—2361no—
273C4⋊C4—2361no—
274C4⋊C4—2362no—
275C4⋊C4—2362no—
276C4⋊C4—2362no—
277C4⋊C4—2362no—
278C22⋊C4—41no—
279C22⋊C4—42no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—2361no—
283C22⋊C4—2362no—
284C22⋊C4—2361no—
285C22⋊C4—2361no—
286C42—1181no—
287C42—1181no—
288C42—1181no—
289C42—1181no—
290C42—2361no—
291C42—2361no—
292C42—2362no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23C22⋊C4×D5911yescharacteristic
298C23#1427375611yescharacteristic
299C23#1427375911yescharacteristic
300Q8—42no—
301Q8—42no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—591no—
316C2×C4—591no—
317C2×C4—1181no—
318C2×C4—1181no—
319C2×C4—1182no—
320C2×C4—1181no—
321C2×C4—1181no—
322C2×C4—1181no—
323C2×C4—1181no—
324C2×C4—1181no—
325C2×C4—4721no—
326C2×C4—4721no—
327C2×C4—4721no—
328C2×C4—2361no—
329C2×C4—2361no—
330C2×C4—2361no—
331C2×C4—2361no—
332C2×C4—2361no—
333C2×C4—2361no—
334C2×C4—2361no—
335C2×C4—2361no—
336C2×C4—2361no—
337C2×C4—2361no—
338C2×C4—2361no—
339C8—81no—
340C8—4721no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—81no—
350C22#99×D5911yescharacteristic
351C22#1683740911yescharacteristic
352C22#1683741711yescharacteristic
353C4—42no—
354C4—41no—
355C4—41no—
356C4—81no—
357C4—161no—
358C4—1181no—
359C4—1181no—
360C4—9441no—
361C4—4721no—
362C4—2362no—
363C4—2361no—
364C4—2361no—
365C4—2361no—
366C2—21no—
367C2—21no—
368C2—41no—
369C2—41no—
370C2#2142262711yescharacteristic
371C2#2142573411yescharacteristic
372C2#2142576011yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1351C2
socleC22×C59C2×#99
prime radicalC24×C59C2×D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1351D59
hypocenterC59#11810
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1351

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
8. C2×Q8×Dic59C4semidirect
11. #16837342C4semidirect
15. #16837321C4semidirect
18. #16837253C4semidirect
213. C59#11810semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214273632
#214241722
#214224272
#214221212
#214221192
#214221172
C59×#13512
#1181059